C₅×#2099 #10603607

order	640
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#2099

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

214 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7,10,12)(8,13,9,11)(14,15)(16,18)(17,19)(20,21);(6,8,10,9)(7,11,12,13)(14,16,17,20)(15,18,19,21)(22,23)(24,26)(25,28)(27,29);(6,9,10,8)(7,11,12,13)(15,19)(18,21)(23,26)(24,29)(27,28)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11988328, #11988335, #11988356, …
maximal subgroup of	C₅×#7787, C₅×#7788, C₅×#7791, …
maximal normal subgroup of	C₅×#7787, C₅×#7788, C₅×#7791, …
maximal quotient of	C₅×#10687, C₅×#11207, C₅×#11267, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 5689);